Non-Parametric Survival

Non-Parametric Survival in Economic Evaluation

Non-parametric survival is the picture of survival that a trial or cohort gives before any model is fitted, usually a Kaplan-Meier curve for each arm drawn as a step function that drops at each observed death or progression. In health economics it is the evidence against which every extrapolated curve is judged, and in some models it supplies the survival probabilities for the observed period directly. It cannot say anything about survival after the last follow-up, which is why most lifetime cost-effectiveness models pair it with a fitted distribution. The estimation methods belong to non-parametric estimation, while for economic evaluation the questions are how the resulting estimates are used in a model and where they stop being enough.

Estimates read directly from the observed data

Survival data from a trial are incomplete because some patients are lost to follow-up and others have not had the event when the study ends. NICE Decision Support Unit Technical Support Document 14 (TSD 14), on survival analysis for economic evaluations alongside clinical trials, notes that these censored patients still provide a lower bound for their unobserved survival time, so survival methods use them rather than excluding them. A non-parametric estimate does this without assuming that survival times follow an exponential, Weibull or any other named distribution.

The result is an empirical survival function. With no censoring, the Kaplan-Meier estimator reduces to the plain proportion of patients still event-free at each time. With censoring, it changes only at observed event times, stays flat at censoring times, and, when the last observation is censored, is undefined beyond the last death. That last property is the practical boundary for economic modelling, since the curve ends where the data end.

The product-limit and cumulative hazard estimates

Two estimators carry most of the work, and both are built from the same risk sets. The product-limit estimator multiplies the conditional probabilities of surviving each distinct event time.

$$\hat{S}(t) = \prod_{i:\, t_{(i)} \le t} \left(1 - \frac{d_i}{n_i}\right)$$

where $\hat{S}(t)$ is the estimated probability of remaining event-free beyond time $t$, $t_{(i)}$ is the $i$th distinct ordered event time, $d_i$ is the number of events at $t_{(i)}$, and $n_i$ is the number of patients still at risk just before $t_{(i)}$, which excludes everyone who has already had the event or been censored.

The Nelson-Aalen estimator adds up the event rates at each event time to estimate the cumulative hazard.

$$\hat{H}(t) = \sum_{i:\, t_{(i)} \le t} \frac{d_i}{n_i}, \qquad \tilde{S}(t) = \exp\{-\hat{H}(t)\}$$

where $\hat{H}(t)$ is the estimated cumulative hazard up to time $t$, $d_i$ and $n_i$ are as above, and $\tilde{S}(t)$ is the survival estimate obtained by exponentiating minus the cumulative hazard, often called the Breslow estimate. The two survival estimates are asymptotically equivalent and usually close when the number of events at each time is small relative to the number at risk. Because $e^{-x} \ge 1 - x$ for every real $x$, the exponentiated Nelson-Aalen curve never sits below the Kaplan-Meier curve.

When the data are grouped into fixed intervals, the life table method gives the actuarial version of the same idea. Groups are compared with the log-rank test, which also makes no distributional assumption.

Where non-parametric survival enters an economic model

An economic model needs mean survival, because costs and QALYs accumulate over the whole time patients spend in each state. TSD 14 recommends that mean time to event be estimated rather than medians, and that parametric models be used rather than restricted means approaches unless the data are almost entirely complete. Non-parametric survival therefore rarely stands alone in a lifetime model, but it appears at four points.

Benchmark for fitted curves

TSD 14 describes visual comparison of a fitted parametric survival model with the Kaplan-Meier curve as a simple way to choose between models, while warning that it is uncertain and should be supplemented with other tests, since a model can follow the curve closely and still have an implausible tail. Where visual inspection is used, the document asks for numbers at risk to be shown on Kaplan-Meier diagrams. Log-cumulative hazard plots, which plot the log of minus the log of the estimated survivor function against log time, use the same non-parametric estimate to check whether Weibull or exponential shapes are plausible and whether hazards look proportional.

Restricted means and the observed-data scenario

The area under a survival curve up to a chosen time is the restricted mean survival time, a measure of average survival between time zero and that point. In the TSD 14 review of 45 NICE technology appraisals, 17 (38%) used a restricted means analysis in the base case or a sensitivity analysis, generally by taking the area under the Kaplan-Meier curve up to the final observation. When censoring is substantial, the document suggests reporting cost-effectiveness based only on observed data as one scenario, because it shows how much of the result depends on the extrapolated period.

Hybrid curves that switch to a parametric tail

Some analyses keep the Kaplan-Meier curve for the observed period and attach a fitted curve afterwards. TSD 14 describes the Liverpool Reviews and Implementation Group approach in TA181, which combined the area under the Kaplan-Meier curve with the area under an exponential model fitted to its tail, and the Gelber method, which the manufacturer in TA118 reported using. It notes that both are sensitive to the point at which the parametric model takes over, so the switch point needs a clear statistical rationale. These approaches resemble a piecewise model with an exponential final segment.

Reconstructing data from published curves

Review groups often lack the individual patient data held by the company. TSD 14 reports that assessment groups typically digitised published Kaplan-Meier curves to estimate patient-level data, and Guyot and colleagues developed an algorithm that inverts the Kaplan-Meier equations to recover a close approximation to the original time-to-event data, using reported numbers at risk and total events where available. Their validation found excellent accuracy for survival probabilities and medians, while hazard ratios were reasonably accurate only when numbers at risk or total events had been reported. The reconstructed data then feed parametric fitting or a partitioned survival model.

Worked example with ten patients and an exponential tail

All numbers in this example are illustrative. Ten patients in one arm are followed for up to 12 months. Deaths occur at months 2, 4, 4, 7 and 9. Patients are censored at months 3, 6 and 10, and two are still alive at the 12-month data cut.

1. Kaplan-Meier and Nelson-Aalen steps. Each row uses the number at risk just before the event time.

At month 4 the risk set is 8 because one patient died at month 2 and one was censored at month 3. At month 7 it is 5 because two died at month 4 and one was censored at month 6. The Kaplan-Meier products are 0.9 × 6/8 = 0.675, 0.675 × 0.8 = 0.54 and 0.54 × 0.75 = 0.405. The exponentiated cumulative hazard is above the Kaplan-Meier value at every step, as expected. The curve first falls to 0.5 or below at month 9, so the median is 9 months.

2. Restricted mean to 12 months. The area under the step function is 2 × 1.00 + 2 × 0.90 + 3 × 0.675 + 2 × 0.54 + 3 × 0.405 = 8.12 months, made up of rectangles of 2.00, 1.80, 2.025, 1.08 and 1.215 months. This is the observed-data mean, and it understates lifetime survival because an estimated 40.5% of the arm is still alive at month 12.

3. Keep the curve to month 12 and add an exponential tail. The tail hazard is estimated from the later part of follow-up. From month 6, five patients remain at risk. They contribute 1 + 3 + 4 + 6 + 6 = 20 person-months up to month 12, with 2 deaths, so the constant hazard is 2 / 20 = 0.10 per month. An exponential curve starting from the Kaplan-Meier value of 0.405 at month 12 has area 0.405 / 0.10 = 4.05 months, so the hybrid mean is 8.12 + 4.05 = 12.17 months. The extrapolated period supplies 4.05 / 12.17 = 0.33, or 33%, of the mean.

4. Move the switch point to month 4. If the exponential replaces the Kaplan-Meier curve from month 4 instead, six patients remain at risk after the two deaths. They contribute 2 + 3 + 5 + 6 + 8 + 8 = 32 person-months up to month 12, with 2 deaths, so the hazard is 2 / 32 = 0.0625 per month. The Kaplan-Meier area to month 4 is 2.00 + 1.80 = 3.80 months and the exponential area from month 4 is 0.675 / 0.0625 = 10.80 months, giving a mean of 14.60 months.

5. Interpretation. The same ten patients give a mean of 8.12, 12.17 or 14.60 months depending on whether and where the curve is extended. With so few patients the highest estimate is 80% above the observed-data mean (14.60 / 8.12 = 1.80), which is the sensitivity TSD 14 warns about. A real analysis would justify the switch point with log-cumulative hazard plots and external data, and would compare it with fully parametric alternatives.

Assumptions behind the estimates

A non-parametric estimate avoids a distributional assumption but still rests on others. TSD 14 states that the standard survival methods it discusses are only suitable if censoring is uninformative, meaning patients who are censored have the same prospects as those who remain under observation. When that fails, as in dependent censoring, the Kaplan-Meier curve is biased however many patients are enrolled.

The timing of assessments matters as well. TSD 14 explains that the stepped shape of a progression-free survival curve can reflect scheduled assessments, every six weeks in its example, rather than the true timing of events, and that interval censoring methods should be considered when the gaps between assessments are long. The definition of the time origin and the event, such as overall survival against progression-free survival, also changes what the curve describes.

Non-parametric, semi-parametric and parametric survival

The three families answer different needs in a model. Non-parametric survival describes what happened in the data and makes no claim beyond the last observation. A semi-parametric model such as the Cox proportional hazards model estimates covariate or treatment effects while leaving the baseline hazard unspecified, so on its own it also bases inferences on the observed data. Parametric survival assumes a distribution, which allows extrapolation and a full mean, at the price of an assumption about the shape of the hazard beyond follow-up.

The families are used together rather than as rivals. The Kaplan-Meier curve is the descriptive summary of the trial, the fitted curve supplies the long-term projection, and the comparison between them is part of the case for the chosen model.

Misreadings and limits

Most problems arise from treating the Kaplan-Meier curve as more certain, or more complete, than it is. The curve is an honest summary of follow-up, and errors creep in when it is asked to carry the long-term projection as well.

1. Reading the right-hand tail as precise. Few patients remain at risk at the end of follow-up, so a single death can move the curve sharply, and the numbers at risk show how much weight the tail can bear.
2. Using the median as the model input. Economic models need means. TSD 14 found medians used in 16 of the 45 appraisals it reviewed, more often in early ones, and reports that assessment groups usually criticised their later use.
3. Trimming the tail before fitting. TSD 14 advises against excluding late data points unless they are shown to be erroneous outliers, calls fitting to one point per month inadvisable, and suggests piecewise modelling instead.
4. Treating the restricted mean as lifetime survival. It is a mean up to the restriction time only.
5. Assuming the hybrid result is assumption-free. The tail model and the switch point are both choices that should be tested in scenario analyses.

Sources

• Guyot P, Ades AE, Ouwens MJNM, Welton NJ. Enhanced secondary analysis of survival data: reconstructing the data from published Kaplan-Meier survival curves. BMC Medical Research Methodology. 2012;12:9. https://doi.org/10.1186/1471-2288-12-9
• Latimer N. NICE DSU Technical Support Document 14: Survival analysis for economic evaluations alongside clinical trials, extrapolation with patient-level data. Sheffield: Decision Support Unit, ScHARR, University of Sheffield; June 2011, last updated March 2013. Sections 1, 3.1, 3.2, 4.1 and 5. https://www.sheffield.ac.uk/sites/default/files/2022-02/TSD14-Survival-analysis.updated-March-2013.v2.pdf
• Rodríguez G. Non-parametric estimation in survival models. Lecture notes, Princeton University; 2001, revised 2005. Sections 1.1 to 1.5. https://grodri.github.io/survival/NonParametricSurvival.pdf
• Royston P, Parmar MKB. Restricted mean survival time: an alternative to the hazard ratio for the design and analysis of randomized trials with a time-to-event outcome. BMC Medical Research Methodology. 2013;13:152. https://doi.org/10.1186/1471-2288-13-152